The Howe duality and polynomial solutions for the symplectic Dirac operator

TL;DR

This paper explores the metaplectic Howe duality and uses Fischer decomposition to identify polynomial solutions of the symplectic Dirac operator, advancing understanding of symplectic monogenics in representation theory.

Contribution

It introduces a detailed analysis of the metaplectic Howe duality and explicitly characterizes symplectic monogenics through Fischer decomposition.

Findings

Determined the space of polynomial solutions to the symplectic Dirac operator.

Connected Howe duality with the structure of symplectic monogenics.

Provided a framework for analyzing polynomial solutions in symplectic representation spaces.

Abstract

We study various aspects of the metaplectic Howe duality realized by Fischer decomposition for the metaplectic representation space of polynomials on $\mathbb{R}^{2n}$ valued in the Segal-Shale-Weil representation. As a consequence, we determine symplectic monogenics, i.e., the space of polynomial solutions of the symplectic Dirac operator.